HINT: <no title>
[−0 points ⇒ 7 / 7 points left]
First choose any two variables to represent the two
different numbers. Then write an equation for the product described in
the question.
STEP: Choose variables for the numbers and then write equations to summarise the information in the question
[−2 points ⇒ 5 / 7 points left]
This question describes facts about two numbers: the numbers have a sum of 20, and we are multiplying one of the numbers by the cube of the other number.
Start by picking symbols for the two numbers. We will use x and y. Now use these variables to write equations:
The sum of the numbers is 20:
One number multiplied by the cube of the other:
We use P to represent the product.
STEP: Rewrite the product in terms of a single variable
[−1 point ⇒ 4 / 7 points left]
Since we want to find a maximum value of P, we will need to find the derivative of the function. However, we cannot find the derivative yet because the expression for P contains two variables, namely x and y.
We need to change it so that it depends on only one variable so that we
can differentiate with respect to that variable: rearrange equation 1 to make x the subject and then substitute this expression into equation 2:
x+y∴x=20=20−y
Substituting:
P=xy3=(20−y)y3=20y3−y4=−y4+20y3
Notice that this is the process for solving
simultaneous equations: we have two equations and we are substituting
one equation into the other to remove one of the variables.
STEP: Differentiate the function
[−1 point ⇒ 3 / 7 points left]
Now we can differentiate the function:
P∴dPdy=−y4+20y3=−4y3+60y2
STEP: Find the value of y which maximises the product and calculate the maximum product
[−1 point ⇒ 2 / 7 points left]
To find the maximum (or minimum) value of the product, we set the derivative of P equal to zero and solve for y.
dPdy(0)=−4y3+60y2=−4y3+60y2
Divide through by a factor of −4:
00y=0=y3−15y2=y2(y−15)ory=15
The solution y=0
corresponds to the minimum value of the product (because anything
multiplied by zero is zero, and we know the product cannot be negative
because the question states that both numbers are positive). Therefore, y=15 corresponds to the maximum product.
STEP: Calculate the maximum value of the product
[−2 points ⇒ 0 / 7 points left]
Now we can calculate the value of the maximum product. First find the value of x which corresponds to this maximum, and then use both values to find the maximum product.
x=20−y=20−(15)=5
Finally:
P∴Pmax=xy3=(5)(15)3=(5)(3375)=16875
Therefore, the maximum value of the product is 16875.
There is a slightly different way to get the answer; once we have the value y=15, we can substitute it into the equation P=−y4+20y3 to get the maximum product. If we solve the question that way, we do not need to find the value of x at the maximum like we did above.
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